Example 4.6 
In this example, we shall be comparing the von-Mises distribution with the wrapped normal distribution. In circular statistics, the von-Mises distribution with scale parameter \(\kappa \)  has been used to approximate the wrapped normal distribution by \(\kappa \) . One question of interest is how big must \(\kappa \)  be in order for the von-Mises distribution to be a good approximation for wrapped normal distribution?
Let \(Z\)  be a wrapped normal distribution with 0 mean and variance \(\sigma ^2\) , denoted \(Z \sim WN(0,\sigma ^2)\)  formulated in the following way [1]}:
Suppose \(U\)  is a distribution on \(\mathbb {R}\) , then the wrapping of \(U\)  denoted \(V\)  is \(V=U\; \mathrm {mod}\; 2\pi \) . If \(U\)  has Lebesgue density \(p_U\) , the Lebesgue density of \(V\)  is
\( p_V(\theta )=\sum _{k=-\infty }^\infty p_U(\theta +2\pi k),\quad \theta \in [-\pi ,\pi ). \) 
